The hydrogen bond is really a special case of dipole forces.
A hydrogen bond is the attractive force between the hydrogen
attached to an electronegative atom of one molecule and an electronegative
atom of a different molecule. Usually the electronegative atom
is oxygen, nitrogen, or fluorine, which has a partial negative
charge. The hydrogen then has the partial positive charge.

To recognize the possibility of hydrogen bonding, examine
the Lewis structure of the molecule. The electronegative atom
must have one or more unshared electron pairs as in the case
of oxygen and nitrogen, and has a negative partial charge. The
hydrogen, which has a partial positive charge tries to find another
atom of oxygen or nitrogen with excess electrons to share and
is attracted to the partial negative charge. This forms the basis
for the hydrogen bond.

In other words - The hydrogen on one molecule attached
to O or N that is attracted to an O or N of a different molecule.

In the graphic on the left, the hydrogen is partially positive
and attracted to the partially negative charge on the oxygen.
Because oxygen has two lone pairs, two different hydrogen bonds
can be made to each oxygen.

This is a very specific bond as indicated. Some combinations
which are not hydrogen bonds include: hydrogen to another hydrogen
or hydrogen to a carbon.

Hydrogen bonding is usually stronger
than normal dipole forces between molecules. Of course hydrogen
bonding is not nearly as strong as normal covalent bonds within
a molecule - it is only about 1/10 as strong. This is still strong
enough to have many important ramifications on the properties
of water.

Comparison of Bond Lengths:

The graphic on the left shows a cluster of water molecules
in the liquid state. Water is a polar molecule, with the oxygen
(red) being the negative area and the hydrogen (white) being
the more positive area. Opposite charges attract.

The bond lengths give some indication of the bond strength.
A normal covalent bond is 0.96 Angstroms, while the hydrogen
bond length is is 1.97 A.

The molecular electrostatic
potential is the potential energy of a proton at a particular
location near a molecule.

Negative electrostatic potential corresponds to a attraction
of the proton by the concentrated electron density in the molecules
(from lone pairs, pi-bonds, etc.) (colored in shades of red).

Positive electrostatic potential corresponds to repulsion
of the proton by the atomic nuclei in regions where low electron
density exists and the nuclear charge is incompletely shielded(colored
in shades of blue).